This is now the official version of the Typeclassopedia and supersedes the version published in the Monad.Reader. Please help update and extend it by editing it yourself or by leaving comments, suggestions, and questions on the talk page.

Abstract

The standard Haskell libraries feature a number of type classes with algebraic or category-theoretic underpinnings. Becoming a fluent Haskell hacker requires intimate familiarity with them all, yet acquiring this familiarity often involves combing through a mountain of tutorials, blog posts, mailing list archives, and IRC logs.

The goal of this document is to serve as a starting point for the student of Haskell wishing to gain a firm grasp of its standard type classes. The essentials of each type class are introduced, with examples, commentary, and extensive references for further reading.

Introduction

Have you ever had any of the following thoughts?

What the heck is a monoid, and how is it different from a monad?

I finally figured out how to use Parsec with do-notation, and someone told me I should use something called Applicative instead. Um, what?

Someone in the #haskell IRC channel used (***), and when I asked lambdabot to tell me its type, it printed out scary gobbledygook that didn’t even fit on one line! Then someone used fmap fmap fmap and my brain exploded.

When I asked how to do something I thought was really complicated, people started typing things like zip.ap fmap.(id &&& wtf) and the scary thing is that they worked! Anyway, I think those people must actually be robots because there’s no way anyone could come up with that in two seconds off the top of their head.

If you have, look no further! You, too, can write and understand concise, elegant, idiomatic Haskell code with the best of them.

There are two keys to an expert Haskell hacker’s wisdom:

Understand the types.

Gain a deep intuition for each type class and its relationship to other type classes, backed up by familiarity with many examples.

It’s impossible to overstate the importance of the first; the patient student of type signatures will uncover many profound secrets. Conversely, anyone ignorant of the types in their code is doomed to eternal uncertainty. “Hmm, it doesn’t compile ... maybe I’ll stick in an
fmap here ... nope, let’s see ... maybe I need another (.) somewhere? ... um ...”

The second key—gaining deep intuition, backed by examples—is also important, but much more difficult to attain. A primary goal of this document is to set you on the road to gaining such intuition. However—

There is no royal road to Haskell. —Euclid

This document can only be a starting point, since good intuition comes from hard work, not from learning the right metaphor. Anyone who reads and understands all of it will still have an arduous journey ahead—but sometimes a good starting point makes a big difference.

It should be noted that this is not a Haskell tutorial; it is assumed that the reader is already familiar with the basics of Haskell, including the standard Prelude, the type system, data types, and type classes.

Solid arrows point from the general to the specific; that is, if there is an arrow from Foo to Bar it means that every Bar is (or should be, or can be made into) a Foo.

Dotted arrows indicate some other sort of relationship.

Monad and ArrowApply are equivalent.

Semigroup, Apply and Comonad are greyed out since they are not actually (yet?) in the standard Haskell libraries ∗.

One more note before we begin. The original spelling of “type class” is with two words, as evidenced by, for example, the Haskell 98 Revised Report, early papers on type classes like Type classes in Haskell and Type classes: exploring the design space, and Hudak et al.’s history of Haskell. However, as often happens with two-word phrases that see a lot of use, it has started to show up as one word (“typeclass”) or, rarely, hyphenated (“type-class”). When wearing my prescriptivist hat, I prefer “type class”, but realize (after changing into my descriptivist hat) that there's probably not much I can do about it.

We now begin with the simplest type class of all: Functor.

Functor

The Functor class (haddock) is the most basic and ubiquitous type class in the Haskell libraries. A simple intuition is that a Functor represents a “container” of some sort, along with the ability to apply a function uniformly to every element in the container. For example, a list is a container of elements, and we can apply a function to every element of a list, using map. As another example, a binary tree is also a container of elements, and it’s not hard to come up with a way to recursively apply a function to every element in a tree.

Another intuition is that a Functor represents some sort of “computational context”. This intuition is generally more useful, but is more difficult to explain, precisely because it is so general. Some examples later should help to clarify the Functor-as-context point of view.

In the end, however, a Functor is simply what it is defined to be; doubtless there are many examples of Functor instances that don’t exactly fit either of the above intuitions. The wise student will focus their attention on definitions and examples, without leaning too heavily on any particular metaphor. Intuition will come, in time, on its own.

Definition

Here is the type class declaration for Functor:

classFunctorfwherefmap::(a->b)->fa->fb

Functor is exported by the Prelude, so no special imports are needed to use it.

First, the f a and f b in the type signature for fmap tell us that f isn’t just a type; it is a type constructor which takes another type as a parameter. (A more precise way to say this is that the kind of f must be * -> *.) For example, Maybe is such a type constructor: Maybe is not a type in and of itself, but requires another type as a parameter, like Maybe Integer. So it would not make sense to say instance Functor Integer, but it could make sense to say instance Functor Maybe.

Now look at the type of fmap: it takes any function from a to b, and a value of type f a, and outputs a value of type f b. From the container point of view, the intention is that fmap applies a function to each element of a container, without altering the structure of the container. From the context point of view, the intention is that fmap applies a function to a value without altering its context. Let’s look at a few specific examples.

Instances

∗ Recall that [] has two meanings in Haskell: it can either stand for the empty list, or, as here, it can represent the list type constructor (pronounced “list-of”). In other words, the type [a] (list-of-a) can also be written [] a.

∗ You might ask why we need a separate map function. Why not just do away with the current list-only map function, and rename fmap to map instead? Well, that’s a good question. The usual argument is that someone just learning Haskell, when using map incorrectly, would much rather see an error about lists than about Functors.

As noted before, the list constructor [] is a functor ∗; we can use the standard list function map to apply a function to each element of a list ∗. The Maybe type constructor is also a functor, representing a container which might hold a single element. The function fmap g has no effect on Nothing (there are no elements to which g can be applied), and simply applies g to the single element inside a Just. Alternatively, under the context interpretation, the list functor represents a context of nondeterministic choice; that is, a list can be thought of as representing a single value which is nondeterministically chosen from among several possibilities (the elements of the list). Likewise, the Maybe functor represents a context with possible failure. These instances are:

instanceFunctor[]wherefmap_[]=[]fmapg(x:xs)=gx:fmapgxs-- or we could just say fmap = mapinstanceFunctorMaybewherefmap_Nothing=Nothingfmapg(Justa)=Just(ga)

As an aside, in idiomatic Haskell code you will often see the letter f used to stand for both an arbitrary Functor and an arbitrary function. In this document, f represents only Functors, and g or h always represent functions, but you should be aware of the potential confusion. In practice, what f stands for should always be clear from the context, by noting whether it is part of a type or part of the code.

There are other Functor instances in the standard libraries; below are a few. Note that some of these instances are not exported by the Prelude; to access them, you can import Control.Monad.Instances.

Either e is an instance of Functor; Either e a represents a container which can contain either a value of type a, or a value of type e (often representing some sort of error condition). It is similar to Maybe in that it represents possible failure, but it can carry some extra information about the failure as well.

((,) e) represents a container which holds an “annotation” of type e along with the actual value it holds. It might be clearer to write it as (e,), by analogy with an operator section like (1+), but that syntax is not allowed in types (although it is allowed in expressions with the TupleSections extension enabled). However, you can certainly think of it as (e,).

((->) e) (which can be thought of as (e ->); see above), the type of functions which take a value of type e as a parameter, is a Functor. As a container, (e -> a) represents a (possibly infinite) set of values of a, indexed by values of e. Alternatively, and more usefully, ((->) e) can be thought of as a context in which a value of type e is available to be consulted in a read-only fashion. This is also why ((->) e) is sometimes referred to as the reader monad; more on this later.

IO is a Functor; a value of type IO a represents a computation producing a value of type a which may have I/O effects. If m computes the value x while producing some I/O effects, then fmap g m will compute the value g x while producing the same I/O effects.

Many standard types from the containers library (such as Tree, Map, and Sequence) are instances of Functor. A notable exception is Set, which cannot be made a Functor in Haskell (although it is certainly a mathematical functor) since it requires an Ord constraint on its elements; fmap must be applicable to any types a and b. However, Set (and other similarly restricted data types) can be made an instance of a suitable generalization of Functor, either by making a and b arguments to the Functor type class themselves, or by adding an associated constraint.

Exercises

Implement Functor instances for Either e and ((->) e).

Implement Functor instances for ((,) e) and for Pair, defined as

dataPaira=Pairaa

Explain their similarities and differences.

Implement a Functor instance for the type ITree, defined as

dataITreea=Leaf(Int->a)|Node[ITreea]

Give an example of a type which cannot be made an instance of Functor (without using undefined).

Is this statement true or false?

The composition of two Functors is also a Functor.

If false, give a counterexample; if true, prove it by exhibiting some appropriate Haskell code.

Laws

As far as the Haskell language itself is concerned, the only requirement to be a Functor is an implementation of fmap with the proper type. Any sensible Functor instance, however, will also satisfy the functor laws, which are part of the definition of a mathematical functor. There are two:

fmapid=idfmap(g.h)=(fmapg).(fmaph)

∗ Technically, these laws make f and fmap together an endofunctor on Hask, the category of Haskell types (ignoring ⊥, which is a party pooper). See Wikibook: Category theory.

Together, these laws ensure that fmap g does not change the structure of a container, only the elements. Equivalently, and more simply, they ensure that fmap g changes a value without altering its context ∗.

The first law says that mapping the identity function over every item in a container has no effect. The second says that mapping a composition of two functions over every item in a container is the same as first mapping one function, and then mapping the other.

As an example, the following code is a “valid” instance of Functor (it typechecks), but it violates the functor laws. Do you see why?

A similar argument also shows that any Functor instance satisfying the first law (fmap id = id) will automatically satisfy the second law as well. Practically, this means that only the first law needs to be checked (usually by a very straightforward induction) to ensure that a Functor instance is valid.

Exercises

Although it is not possible for a Functor instance to satisfy the first Functor law but not the second, the reverse is possible. Give an example of a (bogus) Functor instance which satisfies the second law but not the first.

Which laws are violated by the evil Functor instance for list shown above: both laws, or the first law alone? Give specific counterexamples.

Intuition

There are two fundamental ways to think about fmap. The first has already been mentioned: it takes two parameters, a function and a container, and applies the function “inside” the container, producing a new container. Alternately, we can think of fmap as applying a function to a value in a context (without altering the context).

Just like all other Haskell functions of “more than one parameter”, however, fmap is actually curried: it does not really take two parameters, but takes a single parameter and returns a function. For emphasis, we can write fmap’s type with extra parentheses: fmap :: (a -> b) -> (f a -> f b). Written in this form, it is apparent that fmap transforms a “normal” function (g :: a -> b) into one which operates over containers/contexts (fmap g :: f a -> f b). This transformation is often referred to as a lift; fmap “lifts” a function from the “normal world” into the “f world”.

Further reading

Applicative

A somewhat newer addition to the pantheon of standard Haskell type classes, applicative functors represent an abstraction lying in between Functor and Monad in expressivity, first described by McBride and Paterson. The title of their classic paper, Applicative Programming with Effects, gives a hint at the intended intuition behind the Applicative type class. It encapsulates certain sorts of “effectful” computations in a functionally pure way, and encourages an “applicative” programming style. Exactly what these things mean will be seen later.

Definition

Recall that Functor allows us to lift a “normal” function to a function on computational contexts. But fmap doesn’t allow us to apply a function which is itself in a context to a value in another context. Applicative gives us just such a tool, (<*>). It also provides a method, pure, for embedding values in a default, “effect free” context. Here is the type class declaration for Applicative, as defined in Control.Applicative:

classFunctorf=>Applicativefwherepure::a->fa(<*>)::f(a->b)->fa->fb

Note that every Applicative must also be a Functor. In fact, as we will see, fmap can be implemented using the Applicative methods, so every Applicative is a functor whether we like it or not; the Functor constraint forces us to be honest.

∗ Recall that ($) is just function application: f $ x = f x.

As always, it’s crucial to understand the type signatures. First, consider (<*>): the best way of thinking about it comes from noting that the type of (<*>) is similar to the type of ($)∗, but with everything enclosed in an f. In other words, (<*>) is just function application within a computational context. The type of (<*>) is also very similar to the type of fmap; the only difference is that the first parameter is f (a -> b), a function in a context, instead of a “normal” function (a -> b).

pure takes a value of any type a, and returns a context/container of type f a. The intention is that pure creates some sort of “default” container or “effect free” context. In fact, the behavior of pure is quite constrained by the laws it should satisfy in conjunction with (<*>). Usually, for a given implementation of (<*>) there is only one possible implementation of pure.

(Note that previous versions of the Typeclassopedia explained pure in terms of a type class Pointed, which can still be found in the pointed package. However, the current consensus is that Pointed is not very useful after all. For a more detailed explanation, see Why not Pointed?)

Laws

There are several laws that Applicative instances should satisfy ∗, but only one is crucial to developing intuition, because it specifies how Applicative should relate to Functor (the other four mostly specify the exact sense in which pure deserves its name). This law is:

fmapgx=pureg<*>x

It says that mapping a pure function g over a context x is the same as first injecting g into a context with pure, and then applying it to x with (<*>). In other words, we can decompose fmap into two more atomic operations: injection into a context, and application within a context. The Control.Applicative module also defines (<$>) as a synonym for fmap, so the above law can also be expressed as:

g <$> x = pure g <*> x.

Instances

Most of the standard types which are instances of Functor are also instances of Applicative.

Maybe can easily be made an instance of Applicative; writing such an instance is left as an exercise for the reader.

The list type constructor [] can actually be made an instance of Applicative in two ways; essentially, it comes down to whether we want to think of lists as ordered collections of elements, or as contexts representing multiple results of a nondeterministic computation (see Wadler’s How to replace failure by a list of successes).

Let’s first consider the collection point of view. Since there can only be one instance of a given type class for any particular type, one or both of the list instances of Applicative need to be defined for a newtype wrapper; as it happens, the nondeterministic computation instance is the default, and the collection instance is defined in terms of a newtype called ZipList. This instance is:

To apply a list of functions to a list of inputs with (<*>), we just match up the functions and inputs elementwise, and produce a list of the resulting outputs. In other words, we “zip” the lists together with function application, ($); hence the name ZipList.

The other Applicative instance for lists, based on the nondeterministic computation point of view, is:

instanceApplicative[]wherepurex=[x]gs<*>xs=[gx|g<-gs,x<-xs]

Instead of applying functions to inputs pairwise, we apply each function to all the inputs in turn, and collect all the results in a list.

Now we can write nondeterministic computations in a natural style. To add the numbers 3 and 4 deterministically, we can of course write (+) 3 4. But suppose instead of 3 we have a nondeterministic computation that might result in 2, 3, or 4; then we can write

pure(+)<*>[2,3,4]<*>pure4

or, more idiomatically,

(+)<$>[2,3,4]<*>pure4.

There are several other Applicative instances as well:

IO is an instance of Applicative, and behaves exactly as you would think: to execute m1 <*> m2, first m1 is executed, resulting in a function f, then m2 is executed, resulting in a value x, and finally the value f x is returned as the result of executing m1 <*> m2.

((,) a) is an Applicative, as long as a is an instance of Monoid (section Monoid). The a values are accumulated in parallel with the computation.

The Applicative module defines the Const type constructor; a value of type Const a b simply contains an a. This is an instance of Applicative for any Monoid a; this instance becomes especially useful in conjunction with things like Foldable (section Foldable).

The WrappedMonad and WrappedArrow newtypes make any instances of Monad (section Monad) or Arrow (section Arrow) respectively into instances of Applicative; as we will see when we study those type classes, both are strictly more expressive than Applicative, in the sense that the Applicative methods can be implemented in terms of their methods.

Exercises

Implement an instance of Applicative for Maybe.

Determine the correct definition of pure for the ZipList instance of Applicative—there is only one implementation that satisfies the law relating pure and (<*>).

Intuition

McBride and Paterson’s paper introduces the notation to denote function application in a computational context. If each has type for some applicative functor , and has type , then the entire expression has type . You can think of this as applying a function to multiple “effectful” arguments. In this sense, the double bracket notation is a generalization of fmap, which allows us to apply a function to a single argument in a context.

Why do we need Applicative to implement this generalization of fmap? Suppose we use fmap to apply g to the first parameter x1. Then we get something of type f (t2 -> ... t), but now we are stuck: we can’t apply this function-in-a-context to the next argument with fmap. However, this is precisely what (<*>) allows us to do.

This suggests the proper translation of the idealized notation into Haskell, namely

g<$>x1<*>x2<*>...<*>xn,

recalling that Control.Applicative defines (<$>) as convenient infix shorthand for fmap. This is what is meant by an “applicative style”—effectful computations can still be described in terms of function application; the only difference is that we have to use the special operator (<*>) for application instead of simple juxtaposition.

Note that pure allows embedding “non-effectful” arguments in the middle of an idiomatic application, like

g<$>x1<*>purex2<*>x3

which has type f d, given

g::a->b->c->dx1::fax2::bc3::fc

The double brackets are commonly known as “idiom brackets”, because they allow writing “idiomatic” function application, that is, function application that looks normal but has some special, non-standard meaning (determined by the particular instance of Applicative being used). Idiom brackets are not supported by GHC, but they are supported by the Strathclyde Haskell Enhancement, a preprocessor which (among many other things) translates idiom brackets into standard uses of (<$>) and (<*>). This can result in much more readable code when making heavy use of Applicative.

Further reading

There are many other useful combinators in the standard libraries implemented in terms of pure and (<*>): for example, (*>), (<*), (<**>), (<$), and so on (see haddock for Applicative). Judicious use of such secondary combinators can often make code using Applicatives much easier to read.

McBride and Paterson’s original paper is a treasure-trove of information and examples, as well as some perspectives on the connection between Applicative and category theory. Beginners will find it difficult to make it through the entire paper, but it is extremely well-motivated—even beginners will be able to glean something from reading as far as they are able.

Conal Elliott has been one of the biggest proponents of Applicative. For example, the Pan library for functional images and the reactive library for functional reactive programming (FRP) ∗ make key use of it; his blog also contains many examples of Applicative in action. Building on the work of McBride and Paterson, Elliott also built the TypeCompose library, which embodies the observation (among others) that Applicative types are closed under composition; therefore, Applicative instances can often be automatically derived for complex types built out of simpler ones.

Although the Parsec parsing library (paper) was originally designed for use as a monad, in its most common use cases an Applicative instance can be used to great effect; Bryan O’Sullivan’s blog post is a good starting point. If the extra power provided by Monad isn’t needed, it’s usually a good idea to use Applicative instead.

Monad

It’s a safe bet that if you’re reading this, you’ve heard of monads—although it’s quite possible you’ve never heard of Applicative before, or Arrow, or even Monoid. Why are monads such a big deal in Haskell? There are several reasons.

Haskell does, in fact, single out monads for special attention by making them the framework in which to construct I/O operations.

Haskell also singles out monads for special attention by providing a special syntactic sugar for monadic expressions: the do-notation.

Monad has been around longer than other abstract models of computation such as Applicative or Arrow.

The more monad tutorials there are, the harder people think monads must be, and the more new monad tutorials are written by people who think they finally “get” monads (the monad tutorial fallacy).

I will let you judge for yourself whether these are good reasons.

In the end, despite all the hoopla, Monad is just another type class. Let’s take a look at its definition.

Definition

The Monad type class is exported by the Prelude, along with a few standard instances. However, many utility functions are found in Control.Monad, and there are also several instances (such as ((->) e)) defined in Control.Monad.Instances.

Let’s examine the methods in the Monad class one by one. The type of return should look familiar; it’s the same as pure. Indeed, returnispure, but with an unfortunate name. (Unfortunate, since someone coming from an imperative programming background might think that return is like the C or Java keyword of the same name, when in fact the similarities are minimal.) From a mathematical point of view, every monad is an applicative functor, but for historical reasons, the Monad type class declaration unfortunately does not require this.

We can see that (>>) is a specialized version of (>>=), with a default implementation given. It is only included in the type class declaration so that specific instances of Monad can override the default implementation of (>>) with a more efficient one, if desired. Also, note that although _ >> n = n would be a type-correct implementation of (>>), it would not correspond to the intended semantics: the intention is that m >> n ignores the result of m, but not its effects.

The fail function is an awful hack that has no place in the Monad class; more on this later.

The only really interesting thing to look at—and what makes Monad strictly more powerful than Applicative—is (>>=), which is often called bind. An alternative definition of Monad could look like:

classApplicativem=>Monad'mwhere(>>=)::ma->(a->mb)->mb

We could spend a while talking about the intuition behind (>>=)—and we will. But first, let’s look at some examples.

Instances

Even if you don’t understand the intuition behind the Monad class, you can still create instances of it by just seeing where the types lead you. You may be surprised to find that this actually gets you a long way towards understanding the intuition; at the very least, it will give you some concrete examples to play with as you read more about the Monad class in general. The first few examples are from the standard Prelude; the remaining examples are from the transformers package.

The simplest possible instance of Monad is Identity, which is described in Dan Piponi’s highly recommended blog post on The Trivial Monad. Despite being “trivial”, it is a great introduction to the Monad type class, and contains some good exercises to get your brain working.

The next simplest instance of Monad is Maybe. We already know how to write return/pure for Maybe. So how do we write (>>=)? Well, let’s think about its type. Specializing for Maybe, we have

(>>=)::Maybea->(a->Maybeb)->Maybeb.

If the first argument to (>>=) is Just x, then we have something of type a (namely, x), to which we can apply the second argument—resulting in a Maybe b, which is exactly what we wanted. What if the first argument to (>>=) is Nothing? In that case, we don’t have anything to which we can apply the a -> Maybe b function, so there’s only one thing we can do: yield Nothing. This instance is:

instanceMonadMaybewherereturn=Just(Justx)>>=g=gxNothing>>=_=Nothing

We can already get a bit of intuition as to what is going on here: if we build up a computation by chaining together a bunch of functions with (>>=), as soon as any one of them fails, the entire computation will fail (because Nothing >>= f is Nothing, no matter what f is). The entire computation succeeds only if all the constituent functions individually succeed. So the Maybe monad models computations which may fail.

The Monad instance for the list constructor [] is similar to its Applicative instance; see the exercise below.

Of course, the IO constructor is famously a Monad, but its implementation is somewhat magical, and may in fact differ from compiler to compiler. It is worth emphasizing that the IO monad is the only monad which is magical. It allows us to build up, in an entirely pure way, values representing possibly effectful computations. The special value main, of type IO (), is taken by the runtime and actually executed, producing actual effects. Every other monad is functionally pure, and requires no special compiler support. We often speak of monadic values as “effectful computations”, but this is because some monads allow us to write code as if it has side effects, when in fact the monad is hiding the plumbing which allows these apparent side effects to be implemented in a functionally pure way.

As mentioned earlier, ((->) e) is known as the reader monad, since it describes computations in which a value of type e is available as a read-only environment.
The Control.Monad.Reader module provides the Reader e a type, which is just a convenient newtype wrapper around (e -> a), along with an appropriate Monad instance and some Reader-specific utility functions such as ask (retrieve the environment), asks (retrieve a function of the environment), and local (run a subcomputation under a different environment).

The Control.Monad.Writer module provides the Writer monad, which allows information to be collected as a computation progresses. Writer w a is isomorphic to (a,w), where the output value a is carried along with an annotation or “log” of type w, which must be an instance of Monoid (see section Monoid); the special function tell performs logging.

The Control.Monad.State module provides the State s a type, a newtype wrapper around s -> (a,s). Something of type State s a represents a stateful computation which produces an a but can access and modify the state of type s along the way. The module also provides State-specific utility functions such as get (read the current state), gets (read a function of the current state), put (overwrite the state), and modify (apply a function to the state).

The Control.Monad.Cont module provides the Cont monad, which represents computations in continuation-passing style. It can be used to suspend and resume computations, and to implement non-local transfers of control, co-routines, other complex control structures—all in a functionally pure way. Cont has been called the “mother of all monads” because of its universal properties.

Exercises

Implement a Monad instance for the list constructor, []. Follow the types!

Implement a Monad instance for ((->) e).

Implement Functor and Monad instances for Free f, defined as

dataFreefa=Vara|Node(f(Freefa))

You may assume that f has a Functor instance. This is known as the free monad built from the functor f.

Intuition

Let’s look more closely at the type of (>>=). The basic intuition is that it combines two computations into one larger computation. The first argument, m a, is the first computation. However, it would be boring if the second argument were just an m b; then there would be no way for the computations to interact with one another (actually, this is exactly the situation with Applicative). So, the second argument to (>>=) has type a -> m b: a function of this type, given a result of the first computation, can produce a second computation to be run. In other words, x >>= k is a computation which runs x, and then uses the result(s) of x to decide what computation to run second, using the output of the second computation as the result of the entire computation.

Intuitively, it is this ability to use the output from previous computations to decide what computations to run next that makes Monad more powerful than Applicative. The structure of an Applicative computation is fixed, whereas the structure of a Monad computation can change based on intermediate results.

To see the increased power of Monad from a different point of view, let’s see what happens if we try to implement (>>=) in terms of fmap, pure, and (<*>). We are given a value x of type m a, and a function k of type a -> m b, so the only thing we can do is apply k to x. We can’t apply it directly, of course; we have to use fmap to lift it over the m. But what is the type of fmap k? Well, it’s m a -> m (m b). So after we apply it to x, we are left with something of type m (m b)—but now we are stuck; what we really want is an m b, but there’s no way to get there from here. We can addm’s using pure, but we have no way to collapse multiple m’s into one.

∗ You might hear some people claim that that the definition in terms of return, fmap, and join is the “math definition” and the definition in terms of return and (>>=) is something specific to Haskell. In fact, both alternative definitions were known in the mathematics community long before Haskell picked up monads.

This ability to collapse multiple m’s is exactly the ability provided by the function join :: m (m a) -> m a, and it should come as no surprise that an alternative definition of Monad can be given in terms of join:

classApplicativem=>Monad''mwherejoin::m(ma)->ma

In fact, the earliest definitions of monads in category theory were in terms of return, fmap, and join (often called , , and in the mathematical literature). Haskell uses an alternative formulation with (>>=) instead of join since it is more convenient to use ∗. However, sometimes it can be easier to think about Monad instances in terms of join, since it is a more “atomic” operation. (For example, join for the list monad is just concat.)

Exercises

Implement (>>=) in terms of fmap (or liftM) and join.

Now implement join and fmap (liftM) in terms of (>>=) and return.

Utility functions

The Control.Monad module provides a large number of convenient utility functions, all of which can be implemented in terms of the basic Monad operations (return and (>>=) in particular). We have already seen one of them, namely, join. We also mention some other noteworthy ones here; implementing these utility functions oneself is a good exercise. For a more detailed guide to these functions, with commentary and example code, see Henk-Jan van Tuyl’s tour.

∗ Still, it is unclear how this "bug" should be fixed. Making Monad require a Functor instance has some drawbacks, as mentioned in this 2011 mailing-list discussion. —Geheimdienst

liftM :: Monad m => (a -> b) -> m a -> m b. This should be familiar; of course, it is just fmap. The fact that we have both fmap and liftM is an unfortunate consequence of the fact that the Monad type class does not require a Functor instance, even though mathematically speaking, every monad is a functor. However, fmap and liftM are essentially interchangeable, since it is a bug (in a social rather than technical sense) for any type to be an instance of Monad without also being an instance of Functor∗.

ap :: Monad m => m (a -> b) -> m a -> m b should also be familiar: it is equivalent to (<*>), justifying the claim that the Monad interface is strictly more powerful than Applicative. We can make any Monad into an instance of Applicative by setting pure = return and (<*>) = ap.

sequence :: Monad m => [m a] -> m [a] takes a list of computations and combines them into one computation which collects a list of their results. It is again something of a historical accident that sequence has a Monad constraint, since it can actually be implemented only in terms of Applicative. There is an additional generalization of sequence to structures other than lists, which will be discussed in the section on Traversable.

when :: Monad m => Bool -> m () -> m () conditionally executes a computation, evaluating to its second argument if the test is True, and to return () if the test is False. A collection of other sorts of monadic conditionals can be found in the IfElse package.

mapM :: Monad m => (a -> m b) -> [a] -> m [b] maps its first argument over the second, and sequences the results. The forM function is just mapM with its arguments reversed; it is called forM since it models generalized for loops: the list [a] provides the loop indices, and the function a -> m b specifies the “body” of the loop for each index.

(=<<) :: Monad m => (a -> m b) -> m a -> m b is just (>>=) with its arguments reversed; sometimes this direction is more convenient since it corresponds more closely to function application.

(>=>) :: Monad m => (a -> m b) -> (b -> m c) -> a -> m c is sort of like function composition, but with an extra m on the result type of each function, and the arguments swapped. We’ll have more to say about this operation later. There is also a flipped variant, (<=<).

The guard function is for use with instances of MonadPlus, which is discussed at the end of the Monoid section.

Many of these functions also have “underscored” variants, such as sequence_ and mapM_; these variants throw away the results of the computations passed to them as arguments, using them only for their side effects.

Other monadic functions which are occasionally useful include filterM, zipWithM, foldM, and forever.

Laws

There are several laws that instances of Monad should satisfy (see also the Monad laws wiki page). The standard presentation is:

The first and second laws express the fact that return behaves nicely: if we inject a value a into a monadic context with return, and then bind to k, it is the same as just applying k to a in the first place; if we bind a computation m to return, nothing changes. The third law essentially says that (>>=) is associative, sort of. The last law ensures that fmap and liftM are the same for types which are instances of both Functor and Monad—which, as already noted, should be every instance of Monad.

∗ I like to pronounce this operator “fish”, but that’s probably not the canonical pronunciation ...

However, the presentation of the above laws, especially the third, is marred by the asymmetry of (>>=). It’s hard to look at the laws and see what they’re really saying. I prefer a much more elegant version of the laws, which is formulated in terms of (>=>)∗. Recall that (>=>) “composes” two functions of type a -> m b and b -> m c. You can think of something of type a -> m b (roughly) as a function from a to b which may also have some sort of effect in the context corresponding to m. (>=>) lets us compose these “effectful functions”, and we would like to know what properties (>=>) has. The monad laws reformulated in terms of (>=>) are:

return>=>g=gg>=>return=g(g>=>h)>=>k=g>=>(h>=>k)

∗ As fans of category theory will note, these laws say precisely that functions of type a -> m b are the arrows of a category with (>=>) as composition! Indeed, this is known as the Kleisli category of the monad m. It will come up again when we discuss Arrows.

Ah, much better! The laws simply state that return is the identity of (>=>), and that (>=>) is associative ∗. Working out the equivalence between these two formulations, given the definition g >=> h = \x -> g x >>= h, is left as an exercise.

There is also a formulation of the monad laws in terms of fmap, return, and join; for a discussion of this formulation, see the Haskell wikibook page on category theory.

do notation

Haskell’s special do notation supports an “imperative style” of programming by providing syntactic sugar for chains of monadic expressions. The genesis of the notation lies in realizing that something like a >>= \x -> b >> c >>= \y -> d can be more readably written by putting successive computations on separate lines:

a>>=\x->b>>c>>=\y->d

This emphasizes that the overall computation consists of four computations a, b, c, and d, and that x is bound to the result of a, and y is bound to the result of c (b, c, and d are allowed to refer to x, and d is allowed to refer to y as well). From here it is not hard to imagine a nicer notation:

do{x<-a;b;y<-c;d}

(The curly braces and semicolons may optionally be omitted; the Haskell parser uses layout to determine where they should be inserted.) This discussion should make clear that do notation is just syntactic sugar. In fact, do blocks are recursively translated into monad operations (almost) like this:

A final note on intuition: do notation plays very strongly to the “computational context” point of view rather than the “container” point of view, since the binding notation x <- m is suggestive of “extracting” a single x from m and doing something with it. But m may represent some sort of a container, such as a list or a tree; the meaning of x <- m is entirely dependent on the implementation of (>>=). For example, if m is a list, x <- m actually means that x will take on each value from the list in turn.

MonadFix

The MonadFix class describes monads which support the special fixpoint operation mfix :: (a -> m a) -> m a, which allows the output of monadic computations to be defined via recursion. This is supported in GHC and Hugs by a special “recursive do” notation, mdo. For more information, see Levent Erkök’s thesis, Value Recursion in Monadic Computations.

Further reading

Philip Wadler was the first to propose using monads to structure functional programs. His paper is still a readable introduction to the subject.

One of the quirks of the Monad class and the Haskell type system is that it is not possible to straightforwardly declare Monad instances for types which require a class constraint on their data, even if they are monads from a mathematical point of view. For example, Data.Set requires an Ord constraint on its data, so it cannot be easily made an instance of Monad. A solution to this problem was first described by Eric Kidd, and later made into a library named rmonad by Ganesh Sittampalam and Peter Gavin.

There are many good reasons for eschewing do notation; some have gone so far as to consider it harmful.

For the categorically inclined, monads can be viewed as monoids (From Monoids to Monads) and also as closure operators Triples and Closure. Derek Elkins’s article in issue 13 of the Monad.Reader contains an exposition of the category-theoretic underpinnings of some of the standard Monad instances, such as State and Cont. There is also an alternative way to compose monads, using coproducts, as described by Lüth and Ghani, although this method has not (yet?) seen widespread use.

Monad transformers

One would often like to be able to combine two monads into one: for example, to have stateful, nondeterministic computations (State + []), or computations which may fail and can consult a read-only environment (Maybe + Reader), and so on. Unfortunately, monads do not compose as nicely as applicative functors (yet another reason to use Applicative if you don’t need the full power that Monad provides), but some monads can be combined in certain ways.

Standard monad transformers

The monad transformer library transformers provides a number of monad transformers. Each monad transformer takes an existing monad and adds some new capability or feature.

IdentityT is the identity transformer, which maps a monad to (something isomorphic to) itself.

For example, StateT s Maybe is an instance of Monad; computations of type StateT s Maybe a may fail, and have access to a mutable state of type s. Monad transformers can be multiply stacked. One thing to keep in mind while using monad transformers is that the order of composition matters. For example, when a StateT s Maybe a computation fails, the state ceases being updated (indeed, it simply disappears); on the other hand, the state of a MaybeT (State s) a computation may continue to be modified even after the computation has failed. (This may seem backwards, but it is correct. Monad transformers build composite monads “inside out”; MaybeT (State s) a is isomorphic to s -> (Maybe a, s). Lambdabot has an indispensable @unmtl command which you can use to “unpack” a monad transformer stack in this way.)

Definition and laws

All monad transformers should implement the MonadTrans type class, defined in Control.Monad.Trans.Class:

classMonadTranstwherelift::Monadm=>ma->tma

It allows arbitrary computations in the base monad m to be “lifted” into computations in the transformed monad t m. (Note that type application associates to the left, just like function application, so t m a = (t m) a.)

lift must satisfy the laws

lift.return=returnlift(m>>=f)=liftm>>=(lift.f)

which intuitively state that lift transforms m a computations into t m a computations in a "sensible" way, which sends the return and (>>=) of m to the return and (>>=) of t m.

Exercises

What is the kind of t in the declaration of MonadTrans?

Transformer type classes and "capability" style

∗ The only problem with this scheme is the quadratic number of instances required as the number of standard monad transformers grows—but as the current set of standard monad transformers seems adequate for most common use cases, this may not be that big of a deal.

There are also type classes (provided by the mtl package) for the operations of each transformer. For example, the MonadState type class provides the state-specific methods get and put, allowing you to conveniently use these methods not only with State, but with any monad which is an instance of MonadState—including MaybeT (State s), StateT s (ReaderT r IO), and so on. Similar type classes exist for Reader, Writer, Cont, IO, and others ∗.

These type classes serve two purposes. First, they get rid of (most of) the need for explicitly using lift, giving a type-directed way to automatically determine the right number of calls to lift. Simply writing put will be automatically translated into lift . put, lift . lift . put, or something similar depending on what concrete monad stack you are using.

Second, they give you more flexibility to switch between different concrete monad stacks. For example, if you are writing a state-based algorithm, don't write

foo::StateIntCharfoo=modify(*2)>>return'x'

but rather

foo::MonadStateIntm=>mCharfoo=modify(*2)>>return'x'

Now, if somewhere down the line you realize you need to introduce the possibility of failure, you might switch from State Int to MaybeT (State Int). The type of the first version of foo would need to be modified to reflect this change, but the second version of foo can still be used as-is.

However, this sort of "capability-based" style (e.g. specifying that foo works for any monad with the "state capability") quickly runs into problems when you try to naively scale it up: for example, what if you need to maintain two independent states? A very nice framework for solving this and related problems is described by Schrijvers and Olivera (Monads, zippers and views: virtualizing the monad stack, ICFP 2011) and is implemented in the Monatron package.

Further reading

There are two excellent references on monad transformers. Martin Grabmüller’s Monad Transformers Step by Step is a thorough description, with running examples, of how to use monad transformers to elegantly build up computations with various effects. Cale Gibbard’s article on how to use monad transformers is more practical, describing how to structure code using monad transformers to make writing it as painless as possible. Another good starting place for learning about monad transformers is a blog post by Dan Piponi.

Monoid

A monoid is a set together with a binary operation which
combines elements from . The operator is required to be associative
(that is, , for any
which are elements of ), and there must be some element of
which is the identity with respect to .
(If you are familiar with group theory, a monoid is like a
group without the requirement that inverses exist.) For example, the
natural numbers under addition form a monoid: the sum of any two
natural numbers is a natural number; for any
natural numbers , , and ; and zero is the additive
identity. The integers under multiplication also form a monoid, as do
natural numbers under , Boolean values under conjunction and
disjunction, lists under concatenation, functions from a set to itself
under composition ... Monoids show up all over the place, once you
know to look for them.

Definition

The definition of the Monoid type class (defined in
Data.Monoid; haddock) is:

The mempty value specifies the identity element of the monoid, and mappend
is the binary operation. The default definition for mconcat
“reduces” a list of elements by combining them all with mappend,
using a right fold. It is only in the Monoid class so that specific
instances have the option of providing an alternative, more efficient
implementation; usually, you can safely ignore mconcat when creating
a Monoid instance, since its default definition will work just fine.

The Monoid methods are rather unfortunately named; they are inspired
by the list instance of Monoid, where indeed mempty = [] and mappend = (++), but this is misleading since many
monoids have little to do with appending (see these Comments from OCaml Hacker Brian Hurt on the haskell-cafe mailing list).

Laws

Of course, every Monoid instance should actually be a monoid in the
mathematical sense, which implies these laws:

Instances

There are quite a few interesting Monoid instances defined in Data.Monoid.

[a] is a Monoid, with mempty = [] and mappend = (++). It is not hard to check that (x ++ y) ++ z = x ++ (y ++ z) for any lists x, y, and z, and that the empty list is the identity: [] ++ x = x ++ [] = x.

As noted previously, we can make a monoid out of any numeric type under either addition or multiplication. However, since we can’t have two instances for the same type, Data.Monoid provides two newtype wrappers, Sum and Product, with appropriate Monoid instances.

sum [1..5] and product [1..5]. Nevertheless, these instances are useful in more generalized settings, as we will see in the section Foldable.

Any and All are newtype wrappers providing Monoid instances for Bool (under disjunction and conjunction, respectively).

There are three instances for Maybe: a basic instance which lifts a Monoid instance for a to an instance for Maybe a, and two newtype wrappers First and Last for which mappend selects the first (respectively last) non-Nothing item.

Endo a is a newtype wrapper for functions a -> a, which form a monoid under composition.

There are several ways to “lift” Monoid instances to instances with additional structure. We have already seen that an instance for a can be lifted to an instance for Maybe a. There are also tuple instances: if a and b are instances of Monoid, then so is (a,b), using the monoid operations for a and b in the obvious pairwise manner. Finally, if a is a Monoid, then so is the function type e -> a for any e; in particular, g `mappend` h is the function which applies both g and h to its argument and then combines the results using the underlying Monoid instance for a. This can be quite useful and elegant (see example).

The type Ordering = LT || EQ || GT is a Monoid, defined in such a way that mconcat (zipWith compare xs ys) computes the lexicographic ordering of xs and ys (if xs and ys have the same length). In particular, mempty = EQ, and mappend evaluates to its leftmost non-EQ argument (or EQ if both arguments are EQ). This can be used together with the function instance of Monoid to do some clever things (example).

There are also Monoid instances for several standard data structures in the containers library (haddock), including Map, Set, and Sequence.

Monoid is also used to enable several other type class instances.
As noted previously, we can use Monoid to make ((,) e) an instance of Applicative:

Monoid can be similarly used to make ((,) e) an instance of Monad as well; this is known as the writer monad. As we’ve already seen, Writer and WriterT are a newtype wrapper and transformer for this monad, respectively.

Monoid also plays a key role in the Foldable type class (see section Foldable).

Other monoidal classes: Alternative, MonadPlus, ArrowPlus

The Alternative type class (haddock)
is for Applicative functors which also have
a monoid structure:

The MonadPlus documentation states that it is intended to model
monads which also support “choice and failure”; in addition to the
monoid laws, instances of MonadPlus are expected to satisfy

mzero>>=f=mzerov>>mzero=mzero

which explains the sense in which mzero denotes failure. Since
mzero should be the identity for mplus, the computation m1 `mplus` m2 succeeds (evaluates to something other than mzero) if
either m1 or m2 does; so mplus represents choice. The guard
function can also be used with instances of MonadPlus; it requires a
condition to be satisfied and fails (using mzero) if it is not. A
simple example of a MonadPlus instance is [], which is exactly the
same as the Monoid instance for []: the empty list represents
failure, and list concatenation represents choice. In general,
however, a MonadPlus instance for a type need not be the same as its
Monoid instance; Maybe is an example of such a type. A great
introduction to the MonadPlus type class, with interesting examples
of its use, is Doug Auclair’s MonadPlus: What a Super Monad! in the Monad.Reader issue 11.

There used to be a type class called MonadZero containing only
mzero, representing monads with failure. The do-notation requires
some notion of failure to deal with failing pattern matches.
Unfortunately, MonadZero was scrapped in favor of adding the fail
method to the Monad class. If we are lucky, someday MonadZero will
be restored, and fail will be banished to the bit bucket where it
belongs (see MonadPlus reform proposal). The idea is that any
do-block which uses pattern matching (and hence may fail) would require
a MonadZero constraint; otherwise, only a Monad constraint would be
required.

Further reading

Monoids have gotten a fair bit of attention recently, ultimately due
to
a blog post by Brian Hurt, in which he
complained about the fact that the names of many Haskell type classes
(Monoid in particular) are taken from abstract mathematics. This
resulted in a long haskell-cafe thread
arguing the point and discussing monoids in general.

In a similar vein, David Place’s article on improving Data.Map in
order to compute incremental folds (see the Monad Reader issue 11)
is also a
good example of using Monoid to generalize a data structure.

As unlikely as it sounds, monads can actually be viewed as a sort of
monoid, with join playing the role of the binary operation and
return the role of the identity; see Dan Piponi’s blog post.

Foldable

The Foldable class, defined in the Data.Foldable
module (haddock), abstracts over containers which can be
“folded” into a summary value. This allows such folding operations
to be written in a container-agnostic way.

This may look complicated, but in fact, to make a Foldable instance
you only need to implement one method: your choice of foldMap or
foldr. All the other methods have default implementations in terms
of these, and are presumably included in the class in case more
efficient implementations can be provided.

Instances and examples

The type of foldMap should make it clear what it is supposed to do:
given a way to convert the data in a container into a Monoid (a
function a -> m) and a container of a’s (t a), foldMap
provides a way to iterate over the entire contents of the container,
converting all the a’s to m’s and combining all the m’s with
mappend. The following code shows two examples: a simple
implementation of foldMap for lists, and a binary tree example
provided by the Foldable documentation.

The foldr function has a type similar to the foldr found in the Prelude, but
more general, since the foldr in the Prelude works only on lists.

The Foldable module also provides instances for Maybe and Array;
additionally, many of the data structures found in the standard containers library (for example, Map, Set, Tree,
and Sequence) provide their own Foldable instances.

Derived folds

Given an instance of Foldable, we can write generic,
container-agnostic functions such as:

-- Compute the size of any container.containerSize::Foldablef=>fa->IntcontainerSize=getSum.foldMap(const(Sum1))-- Compute a list of elements of a container satisfying a predicate.filterF::Foldablef=>(a->Bool)->fa->[a]filterFp=foldMap(\a->ifpathen[a]else[])-- Get a list of all the Strings in a container which include the-- letter a.aStrings::Foldablef=>fString->[String]aStrings=filterF(elem'a')

The Foldable module also provides a large number of predefined
folds, many of which are generalized versions of Prelude functions of the
same name that only work on lists: concat, concatMap, and,
or, any, all, sum, product, maximum(By),
minimum(By), elem, notElem, and find. The reader may enjoy
coming up with elegant implementations of these functions using fold
or foldMap and appropriate Monoid instances.

There are also generic functions that work with Applicative or
Monad instances to generate some sort of computation from each
element in a container, and then perform all the side effects from
those computations, discarding the results: traverse_, sequenceA_,
and others. The results must be discarded because the Foldable
class is too weak to specify what to do with them: we cannot, in
general, make an arbitrary Applicative or Monad instance into a
Monoid. If we do have an Applicative or Monad with a monoid
structure—that is, an Alternative or a MonadPlus—then we can
use the asum or msum functions, which can combine the results as
well. Consult the Foldable documentation for
more details on any of these functions.

Note that the Foldable operations always forget the structure of
the container being folded. If we start with a container of type t a for some Foldable t, then t will never appear in the output
type of any operations defined in the Foldable module. Many times
this is exactly what we want, but sometimes we would like to be able
to generically traverse a container while preserving its
structure—and this is exactly what the Traversable class provides,
which will be discussed in the next section.

Further reading

The Foldable class had its genesis in McBride and Paterson’s paper
introducing Applicative, although it has
been fleshed out quite a bit from the form in the paper.

As you can see, every Traversable is also a foldable functor. Like
Foldable, there is a lot in this type class, but making instances is
actually rather easy: one need only implement traverse or
sequenceA; the other methods all have default implementations in
terms of these functions. A good exercise is to figure out what the default
implementations should be: given either traverse or sequenceA, how
would you define the other three methods? (Hint for mapM:
Control.Applicative exports the WrapMonad newtype, which makes any
Monad into an Applicative. The sequence function can be implemented in terms
of mapM.)

Intuition

The key method of the Traversable class, and the source of its
unique power, is sequenceA. Consider its type:

sequenceA::Applicativef=>t(fa)->f(ta)

This answers the fundamental question: when can we commute two
functors? For example, can we turn a tree of lists into a list of
trees? (Answer: yes, in two ways. Figuring out what they are, and
why, is left as an exercise. A much more challenging question is
whether a list of trees can be turned into a tree of lists.)

The ability to compose two monads depends crucially on this ability to
commute functors. Intuitively, if we want to build a composed monad
M a = m (n a) out of monads m and n, then to be able to
implement join :: M (M a) -> M a, that is,
join :: m (n (m (n a))) -> m (n a), we have to be able to commute
the n past the m to get m (m (n (n a))), and then we can use the
joins for m and n to produce something of type m (n a). See
Mark Jones’s paper for more details.

Instances and examples

What’s an example of a Traversable instance?
The following code shows an example instance for the same
Tree type used as an example in the previous Foldable section. It
is instructive to compare this instance with a Functor instance for
Tree, which is also shown.

It should be clear that the Traversable and Functor instances for
Tree are almost identical; the only difference is that the Functor
instance involves normal function application, whereas the
applications in the Traversable instance take place within an
Applicative context, using (<$>) and (<*>). In fact, this will
be
true for any type.

Any Traversable functor is also Foldable, and a Functor. We can see
this not only from the class declaration, but by the fact that we can
implement the methods of both classes given only the Traversable
methods. A good exercise is to implement fmap and foldMap using
only the Traversable methods; the implementations are surprisingly
elegant. The Traversable module provides these
implementations as fmapDefault and foldMapDefault.

The standard libraries provide a number of Traversable instances,
including instances for [], Maybe, Map, Tree, and Sequence.
Notably, Set is not Traversable, although it is Foldable.

Category

Category is another fairly new addition to the Haskell standard
libraries; you may or may not have it installed depending on the
version of your base package. It generalizes the notion of
function composition to general “morphisms”.

The definition of the Category type class (from
Control.Category—haddock) is shown below. For ease of reading, note that I have used an
infix type constructor (~>), much like the infix function type
constructor (->). This syntax is not part of Haskell 98.
The second definition shown is the one used in the standard libraries.
For the remainder of this document, I will use the infix type
constructor (~>) for Category as well as Arrow.

classCategory(~>)whereid::a~>a(.)::(b~>c)->(a~>b)->(a~>c)-- The same thing, with a normal (prefix) type constructorclassCategorycatwhereid::cataa(.)::catbc->catab->catac

Note that an instance of Category should be a type constructor which
takes two type arguments, that is, something of kind * -> * -> *. It
is instructive to imagine the type constructor variable cat replaced
by the function constructor (->): indeed, in this case we recover
precisely the familiar identity function id and function composition
operator (.) defined in the standard Prelude.

Of course, the Category module provides exactly such an instance of
Category for (->). But it also provides one other instance, shown
below, which should be familiar from the
previous discussion of the Monad laws. Kleisli m a b, as defined
in the Control.Arrow module, is just a newtype wrapper around a -> m b.

The only law that Category instances should satisfy is that id and
(.) should form a monoid—that is, id should be the identity of
(.), and (.) should be associative.

Finally, the Category module exports two additional operators:
(<<<), which is just a synonym for (.), and (>>>), which is
(.) with its arguments reversed. (In previous versions of the
libraries, these operators were defined as part of the Arrow class.)

Arrow

The Arrow class represents another abstraction of computation, in a
similar vein to Monad and Applicative. However, unlike Monad
and Applicative, whose types only reflect their output, the type of
an Arrow computation reflects both its input and output. Arrows
generalize functions: if (~>) is an instance of Arrow, a value of
type b ~> c can be thought of as a computation which takes values of
type b as input, and produces values of type c as output. In the
(->) instance of Arrow this is just a pure function; in general, however,
an arrow may represent some sort of “effectful” computation.

Definition

The definition of the Arrow type class, from
Control.Arrow (haddock), is:

∗ In versions of the base
package prior to version 4, there is no Category class, and the
Arrow class includes the arrow composition operator (>>>). It
also includes pure as a synonym for arr, but this was removed
since it conflicts with the pure from Applicative.

The first thing to note is the Category class constraint, which
means that we get identity arrows and arrow composition for free:
given two arrows g :: b ~> c and h :: c ~> d, we can form their
composition g >>> h :: b ~> d∗.

As should be a familiar pattern by now, the only methods which must be
defined when writing a new instance of Arrow are arr and first;
the other methods have default definitions in terms of these, but are
included in the Arrow class so that they can be overridden with more
efficient implementations if desired.

Intuition

The arr function takes any function b -> c and turns it into a generalized arrow b ~> c. The arr method justifies the claim that arrows generalize functions, since it says that we can treat any function as an arrow. It is intended that the arrow arr g is “pure” in the sense that it only computes g and has no “effects” (whatever that might mean for any particular arrow type).

The first method turns any arrow from b to c into an arrow from (b,d) to (c,d). The idea is that first g uses g to process the first element of a tuple, and lets the second element pass through unchanged. For the function instance of Arrow, of course, first g (x,y) = (g x, y).

The second function is similar to first, but with the elements of the tuples swapped. Indeed, it can be defined in terms of first using an auxiliary function swap, defined by swap (x,y) = (y,x).

The (***) operator is “parallel composition” of arrows: it takes two arrows and makes them into one arrow on tuples, which has the behavior of the first arrow on the first element of a tuple, and the behavior of the second arrow on the second element. The mnemonic is that g *** h is the product (hence *) of g and h. For the function instance of Arrow, we define (g *** h) (x,y) = (g x, h y). The default implementation of (***) is in terms of first, second, and sequential arrow composition (>>>). The reader may also wish to think about how to implement first and second in terms of (***).

The (&&&) operator is “fanout composition” of arrows: it takes two arrows g and h and makes them into a new arrow g &&& h which supplies its input as the input to both g and h, returning their results as a tuple. The mnemonic is that g &&& h performs both gandh (hence &) on its input. For functions, we define (g &&& h) x = (g x, h x).

Instances

The Arrow library itself only provides two Arrow instances, both
of which we have already seen: (->), the normal function
constructor, and Kleisli m, which makes functions of
type a -> m b into Arrows for any Monad m. These instances are:

Note that this version of the laws is slightly different than the laws given in the
first two above references, since several of the laws have now been
subsumed by the Category laws (in particular, the requirements that
id is the identity arrow and that (>>>) is associative). The laws
shown here follow those in Paterson’s Programming with Arrows, which uses the
Category class.

∗ Unless category-theory-induced insomnolence is your cup of tea.

The reader is advised not to lose too much sleep over the Arrow
laws ∗, since it is not essential to understand them in order to
program with arrows. There are also laws that ArrowChoice,
ArrowApply, and ArrowLoop instances should satisfy; the interested
reader should consult Paterson: Programming with Arrows.

ArrowChoice

Computations built using the Arrow class, like those built using
the Applicative class, are rather inflexible: the structure of the computation
is fixed at the outset, and there is no ability to choose between
alternate execution paths based on intermediate results.
The ArrowChoice class provides exactly such an ability:

A comparison of ArrowChoice to Arrow will reveal a striking
parallel between left, right, (+++), (|||) and first,
second, (***), (&&&), respectively. Indeed, they are dual:
first, second, (***), and (&&&) all operate on product types
(tuples), and left, right, (+++), and (|||) are the
corresponding operations on sum types. In general, these operations
create arrows whose inputs are tagged with Left or Right, and can
choose how to act based on these tags.

If g is an arrow from b to c, then left g is an arrow from Either b d to Either c d. On inputs tagged with Left, the left g arrow has the behavior of g; on inputs tagged with Right, it behaves as the identity.

The right function, of course, is the mirror image of left. The arrow right g has the behavior of g on inputs tagged with Right.

The (+++) operator performs “multiplexing”: g +++ h behaves as g on inputs tagged with Left, and as h on inputs tagged with Right. The tags are preserved. The (+++) operator is the sum (hence +) of two arrows, just as (***) is the product.

The (|||) operator is “merge” or “fanin”: the arrow g ||| h behaves as g on inputs tagged with Left, and h on inputs tagged with Right, but the tags are discarded (hence, g and h must have the same output type). The mnemonic is that g ||| h performs either gorh on its input.

The ArrowChoice class allows computations to choose among a finite number of execution paths, based on intermediate results. The possible
execution paths must be known in advance, and explicitly assembled with (+++) or (|||). However, sometimes more flexibility is
needed: we would like to be able to compute an arrow from intermediate results, and use this computed arrow to continue the computation. This is the power given to us by ArrowApply.

ArrowApply

The ArrowApply type class is:

classArrow(~>)=>ArrowApply(~>)whereapp::(b~>c,b)~>c

If we have computed an arrow as the output of some previous
computation, then app allows us to apply that arrow to an input,
producing its output as the output of app. As an exercise, the
reader may wish to use app to implement an alternative “curried”
version, app2 :: b ~> ((b ~> c) ~> c).

This notion of being able to compute a new computation
may sound familiar:
this is exactly what the monadic bind operator (>>=) does. It
should not particularly come as a surprise that ArrowApply and
Monad are exactly equivalent in expressive power. In particular,
Kleisli m can be made an instance of ArrowApply, and any instance
of ArrowApply can be made a Monad (via the newtype wrapper
ArrowMonad). As an exercise, the reader may wish to try
implementing these instances:

ArrowLoop

It describes arrows that can use recursion to compute results, and is
used to desugar the rec construct in arrow notation (described
below).

Taken by itself, the type of the loop method does not seem to tell
us much. Its intention, however, is a generalization of the trace
function which is also shown. The d component of the first arrow’s
output is fed back in as its own input. In other words, the arrow
loop g is obtained by recursively “fixing” the second component of
the input to g.

It can be a bit difficult to grok what the trace function is doing.
How can d appear on the left and right sides of the let? Well,
this is Haskell’s laziness at work. There is not space here for a
full explanation; the interested reader is encouraged to study the
standard fix function, and to read Paterson’s arrow tutorial.

Arrow notation

Programming directly with the arrow combinators can be painful,
especially when writing complex computations which need to retain
simultaneous reference to a number of intermediate results. With
nothing but the arrow combinators, such intermediate results must be
kept in nested tuples, and it is up to the programmer to remember
which intermediate results are in which components, and to swap,
reassociate, and generally mangle tuples as necessary. This problem
is solved by the special arrow notation supported by GHC, similar to
do notation for monads, that allows names to be assigned to
intermediate results while building up arrow computations. An example
arrow implemented using arrow notation, taken from
Paterson, is:

Although Hughes’s goal in defining the Arrow class was to
generalize Monads, and it has been said that Arrow lies “between
Applicative and Monad” in power, they are not directly
comparable. The precise relationship remained in some confusion until
analyzed by Lindley, Wadler, and Yallop, who
also invented a new calculus of arrows, based on the lambda calculus,
which considerably simplifies the presentation of the arrow laws
(see The arrow calculus).

Comonad

The final type class we will examine is Comonad. The Comonad class
is the categorical dual of Monad; that is, Comonad is like Monad
but with all the function arrows flipped. It is not actually in the
standard Haskell libraries, but it has seen some interesting uses
recently, so we include it here for completeness.

As you can see, extract is the dual of return, duplicate is the
dual of join, and extend is the dual of (>>=) (although its
arguments are in a different order). The definition
of Comonad is a bit redundant (after all, the Monad class does not
need join), but this is so that a Comonad can be defined by fmap,
extract, and eitherduplicate or extend. Each has a
default implementation in terms of the other.

A prototypical example of a Comonad instance is:

-- Infinite lazy streamsdataStreama=Consa(Streama)instanceFunctorStreamwherefmapg(Consxxs)=Cons(gx)(fmapgxs)instanceCopointedStreamwhereextract(Consx_)=x-- 'duplicate' is like the list function 'tails'-- 'extend' computes a new Stream from an old, where the element-- at position n is computed as a function of everything from-- position n onwards in the old StreaminstanceComonadStreamwhereduplicates@(Consxxs)=Conss(duplicatexs)extendgs@(Consxxs)=Cons(gs)(extendgxs)-- = fmap g (duplicate s)

Colophon

The Typeclassopedia was written by Brent Yorgey and initally published in March 2009. Painstakingly converted to wiki syntax by User:Geheimdienst in November 2011, after asking Brent’s permission.

If something like this tex to wiki syntax conversion ever needs to be done again, here are some vim commands that helped:

%s/\\section{\([^}]*\)}/=\1=/gc

%s/\\subsection{\([^}]*\)}/==\1==/gc

%s/^ *\\item /\r* /gc

%s/---/—/gc

%s/\$\([^$]*\)\$/<math>\1\\ <\/math>/gc Appending “\ ” forces images to be rendered. Otherwise, Mediawiki would go back and forth between one font for short <math> tags, and another more Tex-like font for longer tags (containing more than a few characters)""

%s/|\([^|]*\)|/<code>\1<\/code>/gc

%s/\\dots/.../gc

%s/^\\label{.*$//gc

%s/\\emph{\([^}]*\)}/''\1''/gc

%s/\\term{\([^}]*\)}/''\1''/gc

The biggest issue was taking the academic-paper-style citations and turning them into hyperlinks with an appropriate title and an appropriate target. In most cases there was an obvious thing to do (e.g. online PDFs of the cited papers or Citeseer entries). Sometimes, however, it’s less clear and you might want to check the
original Typeclassopedia PDF
with the
original bibliography file.

To get all the citations into the main text, I first tried processing the source with Tex or Lyx. This didn’t work due to missing unfindable packages, syntax errors, and my general ineptitude with Tex.

I then went for the next best solution, which seemed to be extracting all instances of “\cite{something}” from the source and in that order pulling the referenced entries from the .bib file. This way you can go through the source file and sorted-references file in parallel, copying over what you need, without searching back and forth in the .bib file. I used: